Four fundamental forms for representing rigid body motion, i.e. the rotation-of-vector form, the matrix form, the quaternion form and the screw form, all based on the conventional concept that motion is specified by the movement of a reference point together with an orientation change, are expounded. The inter-relationships among these forms are elucidated. The geometric images of these representation-forms, i.e. the motion mappings, are explained, and an innovative duplex-mapping, a powerful tool for mobility analysis of multi-loop mechanisms and robots, is introduced. An exotic motion representation, based on a concept totally different from the conventional concept, is described. This is a purely geometric representation, a representation by a pair of conjugate curves on respective conjugate surfaces, and is called the conjugation form of motion representation. The conversion from the conjugation form to the rotation-of-vector form and the inverse conversion are described. These conversions, which constitute the essential and featured contents of the Theory of Conjugate Surfaces, have great potential applications in motion design for numerically controlled manufacturing and in surface generation by numerically controlled manufacturing. The basis equations for the conversions, i.e. the three relationship-equations, are deduced. The fundamental equation conjugation, the four conjugato-kinematic entities and the five differential formulas for the inverse conversion are derived.